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Properties

Label 1-33e2-1089.101-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0213i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.625 + 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (0.999 + 0.0190i)7-s + (0.897 + 0.441i)8-s + (−0.415 + 0.909i)10-s + (−0.749 − 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (0.610 − 0.791i)17-s + (0.564 − 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (0.999 + 0.0190i)7-s + (0.897 + 0.441i)8-s + (−0.415 + 0.909i)10-s + (−0.749 − 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (0.610 − 0.791i)17-s + (0.564 − 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0213i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.0213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470797631 + 0.01569976430i\)
\(L(\frac12)\) \(\approx\) \(1.470797631 + 0.01569976430i\)
\(L(1)\) \(\approx\) \(1.034035866 + 0.1427283417i\)
\(L(1)\) \(\approx\) \(1.034035866 + 0.1427283417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.625 + 0.780i)T \)
5 \( 1 + (0.969 - 0.244i)T \)
7 \( 1 + (0.999 + 0.0190i)T \)
13 \( 1 + (-0.749 - 0.662i)T \)
17 \( 1 + (0.610 - 0.791i)T \)
19 \( 1 + (0.564 - 0.825i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (-0.851 + 0.524i)T \)
31 \( 1 + (0.00951 - 0.999i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (0.161 + 0.986i)T \)
43 \( 1 + (-0.928 + 0.371i)T \)
47 \( 1 + (0.710 - 0.703i)T \)
53 \( 1 + (-0.0855 - 0.996i)T \)
59 \( 1 + (0.935 - 0.353i)T \)
61 \( 1 + (-0.988 + 0.151i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (0.921 + 0.389i)T \)
79 \( 1 + (-0.999 + 0.0380i)T \)
83 \( 1 + (-0.290 - 0.956i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.969 - 0.244i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.23372894962803284719342858742, −20.81266223847915885049902362432, −19.96798115204299876371942650176, −18.88546036717417774244355770453, −18.46985911689410693941913088862, −17.615259328598792116764288649968, −16.991429339818178319703688687117, −16.46315874257676094167664003245, −14.94627690059912396089618782835, −14.27941304902190335772583594099, −13.58751109185714886753630122238, −12.49715005950914604660998561728, −11.953834539437263886498175575654, −10.88643007852647907947976517080, −10.3743346020821170587212752746, −9.530166943869692905324956860056, −8.79073471707806298640747500443, −7.85347500323208403580503300437, −7.10504180300113285554792034904, −5.90619726426073505615962815023, −4.90814268837785388161553448246, −3.97189190476525976552621432554, −2.77940227126103004060232947164, −1.92970523499973972606513849998, −1.24788400334347575596532406969, 0.854819416752680491986363161, 1.77536670837234908742724363123, 2.835100330255228428333898341507, 4.60405058380328123379942099496, 5.27831016713149991321982719666, 5.787376280650740430204923919438, 7.07754710270612474935291919253, 7.64446864123190685607938721207, 8.55565200147554866802343662339, 9.51749837172195718432352026204, 9.8795688516172858297101528208, 10.99786382556587345158098490604, 11.71984460467291715268605217244, 13.130073345678312252099046300206, 13.662146414336907474858380656506, 14.69150440015680797258738000271, 14.985068481851152169012712080821, 16.17647464382272624409451446970, 16.90935761215439106870096440857, 17.57127925036037985665941005856, 18.08080074526320512750082184549, 18.77663224289844639091404440274, 20.01148945055971146774352432674, 20.42568217613904914735541474213, 21.485105510198943154476592372970

Graph of the $Z$-function along the critical line